# Singular Values Calculator

Created by Anna Szczepanek, PhD
Reviewed by Rijk de Wet
Last updated: Jan 18, 2024

Finding the singular values of a matrix has never been easier, thanks to Omni's singular values calculator! You'll discover how to find the singular values of a matrix by hand or via the SVD decomposition. We'll also discuss the differences between singular values vs eigenvalues. Ready? Scroll on down!

## What are the singular values of a matrix?

Let A be a m × n matrix. Then A*A is an n × n matrix, where * denotes the transpose or Hermitian conjugation, depending on whether A has real or complex coefficients. The singular values of A the square roots of the eigenvalues of A*A. Since A*A is positive semi-definite, its eigenvalues are non-negative and so taking their square roots poses no problem.

🔎 If you're not yet familiar with the notion of eigenvalues, make sure to take a look at out eigenvalue and eigenvector calculator to get the grasp of this important concept.

Though their definition is not that complicated, finding singular values of a matrix may not be the easiest calculation in the world. The best way to see some examples is to use our singular values calculator. So it's high time we discussed how to find singular values of a matrix using our tool!

## How to use this singular values calculator?

Omni's singular values of a matrix calculator is as user-friendly as can be! You only need to:

1. Pick the matrix's size: the number of rows and the number of columns.
2. Enter the elements of your matrix.
3. The singular values of your matrix will appear at the bottom of out tool. They are listed in decreasing order.
4. By default, our singular values calculator uses 4 decimal places to display the result. To adjust this parameter, click the Advanced mode button.
5. That's it! Experiment with several types of matrices and observe how the singular values change. You'll see that this singular values of a matrix calculator can be an endless source of joy!

💡 The largest singular value gives you the operator norm of your matrix (provided that we consider the Euclidean norm). To learn more about various matrix norms, go to our matrix norm calculator.

## How do I find the singular values of a matrix?

To compute the singular values of a matrix A, follow these steps:

1. Find A', which is the transpose (or Hermitian conjugation) of A;
2. Calculate A' A by applying the standard matrix multiplication;
3. Compute the eigenvalues of the matrix obtained in Step 2.
4. Take the square root of the eigenvalues found in Step 3.
5. Congrats! You've just determined the singular values of your matrix!

💡 In practice, singular values can easily be determined by running the SVD decomposition on a scientific software of your choice, like MATLAB or Python. The singular values are the diagonal elements in one of the matrices returned by the SVD algorithm. If you're interested, the SVD decomposition is thoroughly explained in our SVD calculator.

## Are singular values the same as eigenvalues?

No, singular values and eigenvalues are different concepts in linear algebra. The simplest comparison of singular values vs eigenvalues include the following facts:

• Every matrix (square or rectangular) has singular values. Only square matrices have eigenvalues.
• Singular values are always real and non-negative. Eigenvalues may be negative or complex.

## FAQ

### What are the singular values of a diagonal matrix?

In the case of diagonal matrices, the singular values are simply equal to the absolute values of the diagonal entries.

### What are the singular values of a symmetric matrix?

For a symmetric matrix, the singular values are equal to the absolute values of the eigenvalues. If the symmetric matrix in question is also positive semi-definite, then its singular values and eigenvalues coincide.

Anna Szczepanek, PhD
Matrix size
Number of rows
2
Number of columns
2
A=
 ⌈ a1 a2 ⌉ ⌊ b1 b2 ⌋
Matrix entries
a₁
a₂
b₁
b₂
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